Technology, Context, and Abstraction

At NCTM in April, Dan Meyer was posing some tough questions about math teaching brought up for him by a really cool interactive article by Bret Victor. Something about the article reminded me of a co-teaching experience I’d had in a 5th grade classroom recently, and reflecting on that experience helped me think about how I’d answer Dan’s question, which was something like, “what is the role of math teachers when technology can do what it does in Bret’s interactive article?” It might help to realize Bret Victor is the man behind the Kill Math project.

The Story
So I was teaching 5th grade kids about area and perimeter using this scenario: you have 36 meters of fencing and want to build a rectangular frog pen using all of it. What are some different pens you could make? If each frog needs 1 square meter of space to flourish, how many frogs can your pen designs hold? Which design holds the most?

One traditional model of teaching suggests that what’s hard for students when solving word problems is getting rid of the fluff and decoding the underlying abstract mathematics hidden in the context, and that if the teacher can restate the problem in mathematical language, it will support the students to solve successfully. Here’s what I observed when we used that model:

Students’ Concrete Action

Teachers’ Abstract Response

Student’s Concrete Response

Mention 36 meters of fence

Re-state the idea as “the perimeter is 36 meters”

Ignore the word perimeter, not use any of the teachers’ taught strategies for finding side lengths of a given perimeter.

Use guess and check and drawing pictures to try to find different shaped rectangles that would use 36 feet of fencing; it’s taking a while.

Remind the student of the “hint” that the first step is to “divide it [perimeter] in half. What is half of 36? Can you find two numbers that add to 18?”

The students can, but as soon as the teacher leaves, they start looking for 4 numbers that add to 18 because they look at the picture and remember that rectangles have 4 sides.

Mention that each frog needs one square meter

Ask, “great, what do square meters measure? Area? Yes! Now you need to find the area of each pen you came up with in part 1.”

Ignore the suggestion to find area; give up on the problem; raise their hand to ask for more help. One student tells me, “I know how to find area, but I don’t get what that has to do with how many frogs can fit.”

The next period we tried an alternate model, in which the context was used to elicit the students’ concrete ideas, and the concrete ideas were valued. The teacher helped the students organize their ideas and look for patterns. In short, the teacher avoided abstraction that the students didn’t suggest, while supporting organization, pattern recognition, and referring back to the concrete.

Once we established that when frog farmers say “pen” they mean fenced-in-space-for-keeping-animals-safe, not ink-based-tool-for-writing, there was enough going on in the context that the students had some ideas about how to draw different pens, check if they fit the farmer’s specifications, and how to try to fit the frogs into the pens.

Students’ Concrete Action

Teachers’ Organizing Response

Student’s Concrete Response

Mention 36 meters of fence

Great, that’s one of the requirements the farmer has

Check their guesses against the 36 meters of fence constraint

Use guess and check and drawing pictures to try to find different shaped rectangles that would use 36 feet of fencing; it’s taking a while.

Organize the guesses that worked into a chart with the columns Length and Width

Immediately generate all of the other missing fence shapes that work, and confirm they had them all. No one explicitly mentioned that L + W = 18, but it was clear from the speed of their mental math they were using some version of that pattern.

Mention that each frog needs one square meter

Diagnose student understanding by asking, “how many frogs do you think will fit in one of your pens?”

Make guesses using reasoning that shows they aren’t making sense of the area the frogs take up: 36 frogs or 9 frogs (each square meter uses 4 of the meters of perimeter).

Assume that 36 meters of fencing means 36 frogs will fit in each pen

Invite students to use a drawing to show how many frogs will fit in a pen

Suddenly blurt out, “I can just multiply these! 6 rows and 12 columns of frogs is 72 frogs!” and even “that’s just the area!” One student who filled her 3×15 pen with lots of small squares (over 100) suddenly said, “I did it this way but I wasn’t supposed to. It should be 45 frogs but I drew the boxes too small. All I had to do was multiply.”

My Reflections
As the Common Core points out with the Mathematical Practice “reason abstractly and quantitatively” one piece that’s really at the heart of mathematics is moving among and making links between different representations of quantity (or shape) and relationships among quantities (or shapes), including abstract representations of the quantities and relationships.

Bret Victor’s work is technology that allows more people to make more of those connections, and make them more strongly better, assuming they can make sense of the technological tools. I think our job as teachers, then, is mostly the job of making sense of the tools: why abstract the problem this way? How does the abstraction map to reality? How does it break? And hopefully to prepare some percentage of our students to be the ones to design and improve these tools and their next generation. It’s really exciting to me to look at his tools; I see them as giving more of my students access to having and sharing really powerful ideas, and I see math teachers as having a role in helping students to become people who can use these tools to solve problems and communicate about them confidently.

I am thinking of a metaphor based on how Blogger or WordPress have changed education in writing: every generation since we were writing by carving wood and stone has faced the challenge of how do you make information legible, useful, engaging, and reach lots of people? The more technology we have, it seems like the more people can try to tackle that challenge, and the less time we have to spend identifying who will be our stonemasons or scribes or printmakers or computer coders and training them in the mechanics. The more time we can spend on the creative, interesting, individual tasks of making each piece of content as legible, useful, and engaging as possible. Again, that’s really exciting to me as a teacher — I get to spend time with students thinking about their ideas, their specific piece of writing (or math) and how best to tackle the messy problems of trying to fit what we generally know to specific peoples’ needs. How fun — it certainly requires both general knowledge of what tends to work (e.g. representing change over time on a Cartesian plane with time as the x-axis and other things on the y-axis) and the habits of mind to apply knowledge and push the envelope (e.g. understanding that it makes sense to ask how we can best show the relationship between time and our unknown variable visually).

A big challenge is to think of what this means for classroom teachers right now, as these tools are being invented. Here are some things that feel really true to me:

Put the strategic right in the center of “use appropriate tools strategically” and recognize that what we call “algebra” in school is a tool. When is it strategic to use? Why has it had the impact that it has on the world? What’s worth knowing about it?

Stop telling kids what’s good for them and show them. Trust that quantitative and spatial abstraction is interesting and useful and spend a lot of time generating contexts that show and motivate learning the powerful tools. No kid would ever persevere at piano if they’d never heard music; few would practice scales if they’d never heard for themselves the tricky fast scales hidden inside tough music they’re trying to master.

Figuring out how to assess students’ ability to move among, generate, and compare different representations and abstractions in the service of solving problems. What does it mean to get good at that? What are the 10-20 big ideas across math at all levels that define just what it is to abstract or quantify a situation (eg the real number line and coordinates which map physical space to quantity, which are at the heart of understanding Bret Victor’s car driving tool)?

Being clear and honest that fluency and drill and practice and lecture belong in math class but that in the absence of fitting into big ideas about quantity, space, relationship, and representation they won’t serve our students. The reason they won’t is that the better technology gets the less demand there will be for people who are good at number crunching and symbol manipulation and the more demand there will be for people with heuristics and strategies and big ideas for solving particular problems.

Keep a balance between investigation into pure and applied mathematics. The world needs dreamers and doers in all domains, and they have to start coming from every classroom, not just the demographically college-bound.

My ideal classroom has students working to solve particular problems that I set up for them and using those problems to identify tools they don’t have. If I think they will be able to use them and remember them after hearing them once or twice, I tell them. If I don’t, I set up experience for my students to learn how to re-invent (or invent) them. And then we ask what new questions we generated or if it’s time for me as the expert to define another challenge.

That means needing a deep well-articulated bank of challenges, a sense of their scope and sequence and different paths through them, clear ideas about what mastery means that are aligned with college and business and citizenship demands, and support for effective intervention that supports not just specific tools like using calculators, solving 2-step equations, graphing, and making a guess and check table, but also habits of mind like abstraction and persevering and evaluating for reasonableness.

That’s a big challenge for the designers of curriculum and support material, and one that I think has only sort of been taken up in any really useful way… if it had been, fewer teachers would spend time re-inventing that wheel! I’m excited about the power of online collaboration to help share the materials teachers have invented and reinvented, and really excited about the power of the internet to help us tag, categorize, comment on, critique, and improve new and existing resources.

Comments

Last year was a big year for me to really start tagging and categorizing online resources — there are a lot of great stuff, but the time it takes to peruse them and make them relevant/doable in one’s OWN classroom is always a challenge. As you said, “… if it had been, fewer teachers would spend time re-inventing that wheel!” Exactly.

If you’re reading this comment and wishing you could access what Fawn Nguyen has categorized, wish no longer! Here is a link to her website, where the Links and Resources sections are rich with the things she’s found.

I think giving a hint to divide the perimeter in half is a big mistake. If students come to realize on their own that this is a bit of a shortcut, fine. Otherwise the students are much better off re-inforcing the basic defination and idea of perimeter by dealing with the 36.

Word problems have different purposes at different times. Your students do not seem to understand what area actually means. So, in this case the idea of the word problem is to create understanding. If the students already understood what area means, then this problem would be a way to see how well they can apply their understanding of two concepts. If you are implying that what you call the “traditional approach” does not work so well because the students don’t actually understand what the underlying math means, I agree with you.

Thanks for making the distinction between different uses of word problems! This was an interesting case because the teacher who led the more traditional, abstraction-based approached, was assuming her students did understand the underlying math (in the sense that they had gone over it and solved other area problems before). The problem turned out to diagnose an underlying lack of understanding about area. Once she figured that out, that’s when we switched to an approach that relied on the context to give meaning, rather than looking to abstract from the context to a math concept.

I wonder how context is made use of, and what the value of moving between the concrete and abstract is, when the underlying mathematical abstractions are well understood (e.g. if the students had a robust understanding of area).